#27375613

order	13568
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Splittings

Fitting	C₅₃×#3331 ⋊ C₂
Sylow	C₅₃ ⋊ #19903

Subgroups

	subgroup	G/H
center	C₂²	D₄×C₂²⋊D₅₃
commutator, cosocle	C₄²×C₅₃	C₂⁴
Frattini	C₄²	C₂³×D₅₃
Fitting	C₅₃×#3331	C₂
socle	C₂²×C₅₃	D₄²
prime radical	#20287058	C₂
hypercenter	#3331	D₅₃
hypocenter	C₅₃	#19903
LCS	G ▷ C₄²×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #3331

606 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 85
transitive degree	μ_t(G) = 6784

A minimal faithful permutation representation — 4 generators on 85 points:

(1,2,4,6,8,10,12,14,16,18,20,22,24,26,28,30,32,34,36,38,40,42,44,46,48,50,52,53,51,49,47,45,43,41,39,37,35,33,31,29,27,25,23,21,19,17,15,13,11,9,7,5,3)(70,74)(71,73)(72,77)(75,79)(76,78)(80,81)(82,85)(83,84);(2,3)(4,5)(6,7)(8,9)(10,11)(12,13)(14,15)(16,17)(18,19)(20,21)(22,23)(24,25)(26,27)(28,29)(30,31)(32,33)(34,35)(36,37)(38,39)(40,41)(42,43)(44,45)(46,47)(48,49)(50,51)(52,53)(55,60)(57,65)(59,66)(61,62)(63,69)(67,68)(70,73,76,75)(71,78,79,74)(72,81,83,82)(77,80,84,85);(54,56,58,64)(55,59,62,68)(57,63,65,69)(60,66,61,67)(70,72,76,83)(71,77,79,84)(73,82,75,81)(74,80,78,85);(54,55)(56,59)(57,60)(58,62)(61,65)(63,66)(64,68)(67,69)(70,71,76,79)(72,80,83,85)(73,78,75,74)(77,81,84,82)

Polycyclic presentation:

2^8,53|a^2=g,b^2=gh,c^2=g,e^2=g,f^2=h,b^a=be,c^a=cf,e^a=eg,f^a=fh,c^b=cg,d^b=de,e^b=eg,f^c=fh,i^c=i^52,e^d=eg

Conjugacy classes & character table

797 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 161089);